The Power Rule of Derivatives

The Power Rule of derivatives is an essential formula in differential calculus. Now we shall prove this formula by definition or first principle.

Let us suppose that the function is of the form y = f\left( x \right) = {x^n}, where n is any constant power.

First we take the increment or small change in the function:

\begin{gathered} y + \Delta y = {\left( {x + \Delta x} \right)^n} \\ \Rightarrow \Delta y = {\left( {x + \Delta x} \right)^n} - y \\ \end{gathered}

Putting the value of function y = {x^n} in the above equation, we get

 \Rightarrow \Delta y = {\left( {x + \Delta x} \right)^n} - {x^n}

Taking x common from the above equation, we get

 \Rightarrow \Delta y = {x^n}{\left( {1 + \frac{{\Delta x}}{x}} \right)^n} - {x^n}

Now taking common {x^n}, we get

 \Rightarrow \Delta y = {x^n}\left[ {{{\left( {1 + \frac{{\Delta x}}{x}} \right)}^n} - 1} \right]

Expanding the above expression using binomial series, we get

\begin{gathered} \Rightarrow \Delta y = {x^n}\left[ {1 + n\left( {\frac{{\Delta x}}{x}} \right) + \frac{{n\left( {n - 1} \right)}}{{2!}}{{\left( {\frac{{\Delta x}}{x}} \right)}^2} + \cdots - 1} \right] \\ \Rightarrow \Delta y = {x^n}\left( {\frac{{\Delta x}}{x}} \right)\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \Rightarrow \Delta y = {x^{n - 1}}\Delta x\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \end{gathered}

Dividing both sides by \Delta x, we get

\begin{gathered} \Rightarrow \frac{{\Delta y}}{{\Delta x}} = \frac{{{x^{n - 1}}\Delta x\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right]}}{{\Delta x}} \\ \Rightarrow \frac{{\Delta y}}{{\Delta x}} = {x^{n - 1}}\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \end{gathered}

Taking the limit of both sides as \Delta x \to 0, we have

\begin{gathered} \Rightarrow \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} {x^{n - 1}}\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \Rightarrow \frac{{dy}}{{dx}} = {x^{n - 1}}\left[ {n + 0 + 0 + \cdots } \right] \\ \Rightarrow \frac{{dy}}{{dx}} = n{x^{n - 1}} \\ \Rightarrow \frac{d}{{dx}}{x^n} = n{x^{n - 1}} \\ \end{gathered}

This shows that the derivative of x power n is n{x^{n - 1}}.

Example: Find the derivative of y = f\left( x \right) = 7{x^3} + 2

We have the given function as

y = 7{x^3} + 2

Differentiating with respect to variable x, we get

\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {7{x^3} + 2} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = 7\frac{d}{{dx}}{x^3} + \frac{d}{{dx}}2 \\ \end{gathered}

Now using the power rule and constant function rule, we have

\frac{{dy}}{{dx}} = 21{x^2}